Efficient Parameter Synthesis Using Optimized State Exploration - - PowerPoint PPT Presentation

MeFoSyLoMa December 15th, 2017 Créteil-Université, Créteil

Efficient Parameter Synthesis Using Optimized State Exploration Strategies

Hoang Gia NGUYEN

Joint work with: Étienne André, Laure Petrucci LIPN, Université Paris 13, CNRS, France

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 1 / 28

Outline

Outline

1

Context

2

Parametric Zone Inclusion

3

Exploration Orders for Parametric Zone Inclusion

4

Implementation and Experiments

5

Conclusions

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 2 / 28

Context

Outline

1

Context

2

Parametric Zone Inclusion

3

Exploration Orders for Parametric Zone Inclusion

4

Implementation and Experiments

5

Conclusions

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 3 / 28

Context Parametric Verification of Real-Time Systems

Parametric Verification of Real-Time Systems

Verification techniques used for critical systems, timed systems where a failure or a too late answer can lead to dramatic consequences! such as:

1 Systems incompletely specified: some timing delays may not be known

yet, or may change

2 Verifying system for numerous values of constants requires a very long

time, or even infinite

⇒ Use parameterised techniques, by using parameters instead of constants, then one can check many values at the same time, but also synthesize good valuations of these timing constants

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 4 / 28

Context Parametric Timed Automata (PTA)

Parametric Timed Automata (PTA)

PTA are a formalism to model and verify concurrent real-time systems [Alur et al., 1993] L1 Invariant : x < 5 L2 Invariant : True Guard : x > 1 Reset : x := 0 Timed Automata-TA L1 Invariant : x < p1 L2 Invariant : True Guard : x > p2 Reset : x := 0 PTA x: Clock p1 / p2: Parameters allow to represent unknown values

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 5 / 28

Context Parametric Timed Automata (PTA)

Parametric Timed Automata (PTA)

PTA are a formalism to model and verify concurrent real-time systems [Alur et al., 1993] L1 Invariant : x < p1 L2 Invariant : True Guard : x > p2 Reset : x := 0 PTA L1 x < 5 L2 x > 1 x := 0 With p1 = 5 and p2 = 1 L1 x < 1 L2 x > 5 x := 0 With p1 = 1 and p2 = 5 System Behaviour depends on the values of parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 5 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example

Example: a part of a parameterized version of the FDDI case study of [Herbreteau and Tran, 2015] Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l2 l3 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l2 l3 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Context Parametric Zone Graph (PZG)

Parametric Zone Graph (PZG)

l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Parametric Zone Graph - PZG

Symbolic state: a symbolic state is a pair made of a location, and an attached parametric zone (constraint) Parametric zone: is a set of valuations defined by conjunctions of constraints on clocks and parameters

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 6 / 28

Parametric Zone Inclusion

Outline

1

Context

2

Parametric Zone Inclusion

3

Exploration Orders for Parametric Zone Inclusion

4

Implementation and Experiments

5

Conclusions

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 7 / 28

Parametric Zone Inclusion Objective

Objective

Problem: the order in which we select the states has a huge impact on the efficiency Goal of this work: perform reachability synthesis, i.e., find valuations for which a given location is reachable; to do this, we use the parametric zone graph → Find efficient exploration order strategies

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 8 / 28

Parametric Zone Inclusion Objective

Objective (cont.)

2 popular exploration orders for model checking algorithms

1 Depth-first search - DFS 2 Breadth-first search - BFS

Many authors (e. g., [Behrmann et al., 2000, Behrmann, 2005]) showed that using BFS is much more efficient than DFS for checking reachability properties in TAs ⇒ modify and optimize the breadth-first search (BFS)

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 9 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True Without parametric zone inclusion s0 l0 True With parametric zone inclusion

Parametric zone inclusion: is an optimization technique relying on the parametric zone graph to speed up the parametric model checking

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 Without parametric zone inclusion s0 l0 True s1 l2 y > 2p1 With parametric zone inclusion

Parametric zone inclusion: is an optimization technique relying on the parametric zone graph to speed up the parametric model checking

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True Without parametric zone inclusion s0 l0 True s1 l2 y > 2p1 s2 l1 True With parametric zone inclusion

Parametric zone inclusion: is an optimization technique relying on the parametric zone graph to speed up the parametric model checking

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l3 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 Without parametric zone inclusion s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 With parametric zone inclusion

Parametric zone inclusion: is an optimization technique relying on the parametric zone graph to speed up the parametric model checking

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True Without parametric zone inclusion s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True With parametric zone inclusion

Parametric zone inclusion: is an optimization technique relying on the parametric zone graph to speed up the parametric model checking

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True Without parametric zone inclusion s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True ⊆ With parametric zone inclusion

Parametric zone including: given two reachable states s1 = (l1, C1) and s2 = (l2, C2), whenever l1 = l2 and C1 ⊆ C2, it is safe to replace s1 with s2 in the analysis

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True Without parametric zone inclusion s0 l0 True s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True With parametric zone inclusion

Parametric zone including: given two reachable states s1 = (l1, C1) and s2 = (l2, C2), whenever l1 = l2 and C1 ⊆ C2, it is safe to replace s1 with s2 in the analysis

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l3 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion s0 l0 True s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 With parametric zone inclusion

Parametric zone including: given two reachable states s1 = (l1, C1) and s2 = (l2, C2), whenever l1 = l2 and C1 ⊆ C2, it is safe to replace s1 with s2 in the analysis

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l3 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion s0 l0 True s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 ⊆ With parametric zone inclusion

Parametric zone including: given two reachable states s1 = (l1, C1) and s2 = (l2, C2), whenever l1 = l2 and C1 ⊆ C2, it is safe to replace s1 with s2 in the analysis

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion Order: s0 → s5 States: 6 s0 l0 True s2 l1 True s4 l2 True s5 l3 y ≤ p2 With parametric zone inclusion Order: s0 → s5 States: 4

Problem: inefficient phenomenon happen is when a larger zone is explored after exploring smaller zones (red states)

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 10 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion Order: s0 → s5 States: 6 s0 l0 True s2 l1 True s4 l2 True s5 l3 y ≤ p2 With parametric zone inclusion Order: s0 → s5 States: 4 s0 l0 True Ideal exploration order

Question: how to reduce inefficient phenomenon or useless computation? → Find an exploration order to explore the biggest zone first!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 11 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion Order: s0 → s5 States: 6 s0 l0 True s2 l1 True s4 l2 True s5 l3 y ≤ p2 With parametric zone inclusion Order: s0 → s5 States: 4 s0 l0 True s1 l1 True Ideal exploration order

Question: how to reduce inefficient phenomenon or useless computation? → Find an exploration order to explore the biggest zone first!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 11 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion Order: s0 → s5 States: 6 s0 l0 True s2 l1 True s4 l2 True s5 l3 y ≤ p2 With parametric zone inclusion Order: s0 → s5 States: 4 s0 l0 True s1 l1 True s2 l2 True Ideal exploration order

Question: how to reduce inefficient phenomenon or useless computation? → Find an exploration order to explore the biggest zone first!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 11 / 28

Parametric Zone Inclusion Parametric Zone Inclusion Illustration

Parametric Zone Inclusion Illustration

s0 l0 True s1 l2 y > 2p1 s2 l1 True s3 l3 2p1 < y ≤ p2 s4 l2 True s5 l3 y ≤ p2 Without parametric zone inclusion Order: s0 → s5 States: 6 s0 l0 True s2 l1 True s4 l2 True s5 l3 y ≤ p2 With parametric zone inclusion Order: s0 → s5 States: 4 s0 l0 True s1 l1 True s2 l2 True s3 l3 y ≤ p2 Ideal exploration order Order: s0 → s3 States: 4

Question: how to reduce inefficient phenomenon or useless computation? → Find an exploration order to explore the biggest zone first!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 11 / 28

Exploration Orders for Parametric Zone Inclusion

Outline

1

Context

2

Parametric Zone Inclusion

3

Exploration Orders for Parametric Zone Inclusion

4

Implementation and Experiments

5

Conclusions

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 12 / 28

Exploration Orders for Parametric Zone Inclusion Exploration Orders Introduction

Exploration Orders Introduction

Our contribution: 2 new exploration orders for PTAs

1 Parametric Ranking Strategy

This strategy assigns a priority value to each state, then it explores the state with highest priority first Inspired by the “ranking system” strategy [Herbreteau and Tran, 2015].

2 Parametric Priority Strategy

A new strategy using an insertion mechanism within an ordered list

• f parametric zones

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 13 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

The main idea: Explore the state having the highest rank Ranking:

1 A new explored state starts with rank infinity (if its constraint is

True) or zero (otherwise)

2 The rank of the larger parametric zone is set higher than the

highest rank of the small parametric zone and those in its subtree (with the same location)

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 14 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l1 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 True rank: ∞ PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 True rank: ∞ s3 l2 True rank: ∞ PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 True rank: ∞ s3 l2 True rank: ∞ ⊆ PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l1 l2 l2 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s2 l1 True rank: ∞ s3 l2 True rank: ∞ PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Parametric Ranking Strategy

l0 l1 l2 l3 l3 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s2 l1 True rank: ∞ s3 l2 True rank: ∞ s4 l3 y ≤ p2 rank: 0 PZG with parametric ranking strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 15 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l3 y ≤ p2 y > 2p1 A PTA example l0 l1 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example

There is no likely improvement if there are no True zones in a model, compared to using the BFS exploration order

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 16 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l0 l1 l2 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l1 l2 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 y > p1 rank: 0 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l3 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 s4 l2 y > p1 rank: 0 + 1 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 s4 l2 y > p1 rank: 0 + 1 ⊆ Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 s4 l2 y > p1 rank: 0 + 1 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l3 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 s4 l2 y > p1 rank: 0 + 1 s5 l3 p1 < y ≤ p2 rank: 0 + 1 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l3 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 s4 l2 y > p1 rank: 0 + 1 s5 l3 p1 < y ≤ p2 rank: 0 + 1 ⊆ Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Ranking Strategy

Drawback of Parametric Ranking Strategy

l0 l1 l2 l3 l3 y > p1 y ≤ p2 y > 2p1 A PTA example s0 l0 True rank: ∞ s2 l1 y > p1 rank: 0 s4 l2 y > p1 rank: 0 + 1 s5 l3 p1 < y ≤ p2 rank: 0 + 1 Parametric ranking strategy

Different zone sizes are assigned with zero rank Inefficient phenomenon detected late!

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 17 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

The main idea: A new explored state is inserted into an ordered waiting list W by ascending zone size, then the state at the head of the list will be explored first The waiting list W structure:

1 Two main parts in W 1

The first (at the head) is the true zones part

2

The other is the non-true zone part composed of several parts each containing ordered comparable zones

True True ... x < 2p1 x < p1 ... x ≥ p2 x ≥ 2p2

• True zones
• non-True zones 1
• non-True zones n

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 18 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l0 l1 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True W: s0 Parametric priority strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 19 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True s1 l2 y > 2p1 W: s1 Parametric priority strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 19 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l1 l1 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 y > p1 W: s2, s1 Parametric priority strategy

Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 19 / 28

Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 y > p1 s3 l2 y > p1 W: s3, s1 Parametric priority strategy

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Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True s1 l2 y > 2p1 s2 l1 y > p1 s3 l2 y > p1 W: s3, s1 ⊆ Parametric priority strategy

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Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l1 l2 l2 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True s2 l1 y > p1 s3 l2 y > p1 W: s3 Parametric priority strategy

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Exploration Orders for Parametric Zone Inclusion Parametric Priority Strategy

Parametric Priority Strategy

l0 l1 l2 l3 l3 y > p1 y ≤ p2 y > 2p1 Our PTA example s0 l0 True s2 l1 y > p1 s3 l2 y > p1 s4 l3 p1 < y ≤ p2 W: s4 Parametric priority strategy

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Exploration Orders for Parametric Zone Inclusion Strategies Comparison

Strategies Comparison

s0 l0 True rank: ∞ s1 l2 y > 2p1 rank: 0 s2 l1 y > p1 rank: 0 s3 l3 2p1 < y ≤ p2 rank: 0 s4 l2 y > p1 rank: 0 + 1 s5 l3 p1 < y ≤ p2 rank: 0 + 1 ⊆ ⊆ Parametric ranking strategy s0 l0 True s1 l2 y > 2p1 s2 l1 y > p1 s3 l2 y > p1 s4 l3 p1 < y ≤ p2 ⊆ Parametric priority strategy

⇒ Parametric priority strategy has less inefficient phenomenon

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Implementation and Experiments

Outline

1

Context

2

Parametric Zone Inclusion

3

Exploration Orders for Parametric Zone Inclusion

4

Implementation and Experiments

5

Conclusions

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Implementation and Experiments

Implementation

Implementation in IMITATOR [André, Fribourg, Kühne, Soulat, 2012]

1

A software tool for parametric verification and robustness analysis

• f real-time systems

Thanks to the Parma Polyhedra Library (PPL) library for solving linear inequality systems

1http://www.imitator.fr/ Hoang Gia NGUYEN (Paris 13) State Exploration Optimization December 15th, 2017 22 / 28

Implementation and Experiments

Experiments

Search orders:

BFS: Traditional breadth-first search LayerBFS: Layer breadth-first search is an extension of breadth-first search BFS, which explores states layer by layer (same depth in the parametric zone graph") RS: BFS with Parametric ranking strategy PRIOR: BFS with Parametric priority strategy

Semi-algorithms for reachability synthesis:

EFsynth (exact synthesis): EF-synthesis problem, “find all parameter valuations for which a given location is reachable” EFc-ex (partial synthesis) : EF-counter-example synthesis problem, “find at least some parameter valuations for which a given location is reachable, and stop as soon as some valuations are found”

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Implementation and Experiments

Experiments for Exact Synthesis EFsynth

EFsynth Existing Search Orders Our Contribution Benchmark Models LayerBFS incl (s) BFS incl (s) RS incl (s) PRIOR incl (s) AndOr 2.512 2.41 1.708 1.714 flipflop-P 121.108 102.42 139.822 140.193 BRP 377.913 370.74 174.038 160.079 Thales-3 627.956 759.987 636.823 597.57 Sched2.100.2 148.169 T.O 249.373 259.895 Sched2.50.2 28.137 217.399 36.81 35.26 FDDI-4 1.315 1.1 1.455 1.285 Fischer-3 0.521 0.48 1.172 1.316 Lynch-5 7.359 7.817 8.859 7.867 F4 21.813 37.558 108.629 96.983 Pipeline-KP12-3-3 T.O T.O T.O T.O RCP 1.105 1.099 0.093 0.095 spsmall 10.132 9.595 11.114 10.232 critical-region-4 T.O T.O T.O T.O blowup 31.635 1.345 1.493 1.134 Normalized Average 3.47236 3.7417 2.85594 2.81208

: best time : 2nd best time T.O: time out (3600s)

RS and PRIOR are slightly faster in EFsynth

Additional experiments with merging and bidirectional inclusion: see paper

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Implementation and Experiments

Experiments for Partial Synthesis EFc-ex

EFc-ex Existing Search Orders Our Contribution Benchmark Models LayerBFS incl (s) BFS incl (s) RS incl (s) PRIOR incl (s) AndOr 0.012 0.011 0.008 0.008 flipflop-P 0.061 0.059 0.029 0.028 BRP 2.874 2.944 0.198 0.188 Thales-3 16.638 19.968 0.237 0.232 Sched2.100.2 0.008 0.004 0.005 0.004 Sched2.50.2 0.028 0.023 0.016 0.015 FDDI-4 0.377 0.291 0.091 0.078 Fischer-3 0.097 0.097 0.057 0.059 Lynch-5 7.408 7.912 8.847 7.829 F4 4.086 6.543 0.364 0.311 Pipeline-KP12-3-3 21.927 18.229 0.042 0.042 RCP 0.51 0.454 0.024 0.02 spsmall 5.862 6.242 0.143 0.143 critical-region-4 1.008 0.821 0.044 0.043 blowup 32.893 1.346 1.337 1.003 Normalized Average 6.03173 5.28516 1.13675 1.06026

: best time : 2nd best time T.O: time out (3600s)

RS and PRIOR dominate other algorithms in EFc-ex

Additional experiments with merging and bidirectional inclusion: see paper

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Implementation and Experiments

Experiment Summary

EFsynth EFc-ex 1 2 3 4 5 6 Normalized average run time (s) Lower is better LayerBFS incl BFS incl RS incl PRIOR incl

RS and PRIOR are better in general

Additional experiments with merging and bidirectional inclusion: see paper

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Conclusions

Outline

1

Context

2

Parametric Zone Inclusion

3

Exploration Orders for Parametric Zone Inclusion

4

Implementation and Experiments

5

Conclusions

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Conclusions

Conclusions

Contributions: Proposed two new exploration order strategies for the parameter synthesis problems Implemented and evaluated in IMITATOR Give an overview of the impact of exploration orders in different parameter synthesis problems. Future work: The waiting strategy of [Herbreteau and Tran, 2015] and the exact acceleration technique [Hendriks and Larsen, 2002] could serve as a basis for future parametric strategies Taking advantage of recent multi-core technology for DFS, by adapting the non-parametric algorithm of [Laarman et al., 2013]

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Bibliography

Bibliography

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Bibliography

References I

Alur, R., Henzinger, T. A., and Vardi, M. Y. (1993). Parametric real-time reasoning. In STOC, pages 592–601. ACM. André, É., Fribourg, L., Kühne, U., and Soulat, R. (2012). IMITATOR 2.5: A tool for analyzing robustness in scheduling problems. In FM, volume 7436 of Lecture Notes in Computer Science. Springer. Behrmann, G. (2005). Distributed reachability analysis in timed automata. STTT, 7(1):19–30. Behrmann, G., Hune, T., and Vaandrager, F. W. (2000). Distributing timed model checking – how the search order matters. In CAV, volume 1855 of Lecture Notes in Computer Science, pages 216–231. Springer. Hendriks, M. and Larsen, K. G. (2002). Exact acceleration of real-time model checking.

• Electr. Notes Theor. Comput. Sci., 65(6):120–139.

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Bibliography

References II

Herbreteau, F. and Tran, T. (2015). Improving search order for reachability testing in timed automata. In FORMATS, volume 9268 of Lecture Notes in Computer Science, pages 124–139. Springer. Laarman, A., Olesen, M. C., Dalsgaard, A. E., Larsen, K. G., and Van De Pol, J. (2013). Multi-core emptiness checking of timed Büchi automata using inclusion abstraction. In CAV, volume 8044 of Lecture Notes in Computer Science, pages 968–983. Springer.

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Licensing

Licensing

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Licensing

Source of the graphics used I

Title: Ocaml logo Author: Amir Chaudhry Source: https://commons.wikimedia.org/wiki/File:Smiley_green_alien_big_eyes.svg License: CC BY-SA 4.0 Title: IMITATOR logo (Typing Monkey) Author: Kater Begemot Source: https://commons.wikimedia.org/wiki/File:Smiley_green_alien_big_eyes.svg License: CC BY-SA 3.0 Title: PPL logo Author: Unknown Source: http://bugseng.com/files/ext/images/site/ppl_mm_8.png License: GCC

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